Brief paperCombined economic and regulatory predictive control☆
Introduction
Economic Model Predictive Control (economic MPC) has emerged as a possible core component of the next-generation process control framework. Economic MPC reconciles real-time economic process optimization with process control, solving optimization problems with general economic cost functions. Observed economic benefits of economic MPC over regulatory MPC are now well documented (Amrit, 2011, Rawlings et al., 2012); however, solutions to challenges such as stability, guaranteed closed-loop economic performance and computational considerations are still at an early stage (Angeli et al., 2012, Diehl et al., 2011, Müller et al., 2013). For general economic cost functions, economic MPC retains the property of average asymptotic performance being at least as good as the best admissible steady-state (Angeli, Amrit, & Rawlings, 2009); however, stability is not necessarily guaranteed (Rawlings et al., 2012). To establish asymptotic stability in the context of economic MPC, one requires the assumption of strict dissipativity to hold (Angeli et al., 2012). It has been observed, through simulation, that convergence of economic MPC is attainable if one extends the economic cost function with a sufficiently convex term (Angeli, Amrit, & Rawlings, 2011). In the context of dissipativity theory, the addition of a sufficiently convex term transforms the system to be strictly dissipative and asymptotic stability of the origin can be concluded (Angeli et al., 2012). The extension of an economic cost function with some convex cost term in the work of Angeli et al. (2011), Angeli et al. (2012) draws close parallels to combined economic and regulatory MPC (Maree & Imsland, 2011), also referred to as dual-objective MPC in the present work. In Maree and Imsland (2011), a weight is incorporated to combine the respective objectives in a convex fashion. Specifying, however, the exact nature of such a weight, such that stability and good closed-loop performance are ensured, is not immediately clear (Rawlings et al., 2012).
Maree and Imsland (2014) proposed sufficient conditions for a dynamic weight function, which if satisfied in the context of dual-objective MPC, promote increased economic performance during transients, and simultaneously ensure that the origin is asymptotically stable under the receding horizon control law. The present work relaxes the conditions required for stability in the aforementioned work. Instead, less-restrictive conditions are proposed, which if satisfied, admits convergence of closed-loop trajectories towards some admissible steady-state. The latter property, in the context of economic MPC, can be interpreted as a safe-park property (Mhaskar, Liu, & Christofides, 2013), and may be desirable during uncertain process conditions or severe disturbances that necessitates conservative process regulation in the vicinity of some admissible steady-state. Additionally, we give a theoretical link between the conditions on the dynamic weight function and the work of Angeli et al. (2012) on the role of dissipativity in economic MPC.
Theoretical concepts are illustrated for an acetylene hydrogenation case. The strength of the proposed dual-objective MPC strategy, with dynamic weighting, is simplicity of numerical implementation, with computational complexity comparable with online optimization problems of same type as nonlinear MPC formulations, with the same number of variables.
Notation
The symbols and define integer and real numbers, respectively. A continuous function belongs to a class- function if and is strictly increasing. A continuous function belongs to a class- (is positive definite) if and . A continuous function belongs to a class- function if , and is nonincreasing in , satisfying . The Euclidean norm for a vector argument is defined as .
Section snippets
Preliminaries
We consider the nonlinear, discrete-time system model with state, , and control input . We assume that (1) is continuous in . Define the mixed constraint set for a compact set . Let be a continuous economic stage cost function. Then, the optimal solution to the steady-state optimization problem defines the economic steady-state, . The existence of follows from the continuity of
Combined regulatory and economic predictive control
Consider, in this section, any general continuous weight function, . A specific weight function, fulfilling the assumptions made here and in Section 4, will be proposed in Section 5. The combined regulatory and economic stage cost function (henceforth referred to as the dual-objective stage cost function) is defined We also define the terminal set being a subset of containing a neighbourhood of the economic optimal steady-state, . In
Stability and convergence analysis
In this section we concern ourselves with analysing the stability and convergence of (7) with a terminal state constraint region being defined , in which . The special case , in which , is omitted for brevity; however, the reader is referred to Maree (2014) for a rigorous analysis. Two assumptions to the former definition of (in the context ofAssumption 1 are proposed, which if satisfied, enable us to either conclude on asymptotic stability of, or the
Economic optimal weight functions
Section 4 stipulated necessary assumptions for a dual-objective stage cost function , which if satisfied, guarantees the economic optimal steady-state point to be asymptotically stable for the closed-loop trajectories (8) with region of attraction . Under relaxed conditions, one is still able to show convergence of trajectories (8) to . In the context of optimizing for economic performance, it is desirable to choose in (4) as close to one as possible. We present a strategy in
Numerical example
Acetylene hydrogenation possesses stoichiometry which may be characterized as consecutive–competitive process reactions. If these reactions are conducted in an isothermal catalytic continuous stirred tank reactor, then the dynamic behaviour of the process can be described by following dimensionless form (Lee & Bailey, 1980): We define and as the concentration of acetylene and hydrogen in the feed, respectively. State is
A link to dissipativity
Asymptotic stability of , in the context of economic MPC, can be shown if the nonlinear system (1) is strictly dissipative with respect to an economic supply rate, (Angeli et al., 2012). If this holds, then we can show, in the context of dual-objective MPC, that (1) will also be strictly dissipative with respect to the dual-objective supply rate, , given is chosen according to Proposition 1. Such a result allows, under the dissipativity assumption, to conclude on
Concluding remarks
The present work proposed a combined dual-objective MPC framework in which economic and regulatory objectives are combined in a convex fashion making use of a dynamic weight function. Special conditions, explicitly embedded in the dynamic weight function, ensure that benefits of both economic MPC and regulatory MPC are observed during the online implementation of the resulting dual-objective receding horizon control law. Economic performance is optimized during process transients, with
Johannes Philippus Maree received Ph.D. degree in Engineering Cybernetics from the Department of Engineering Cybernetics at the Norwegian University of Science and Technology (NTNU) in Trondheim, in 2014. As a part of his Ph.D. studies, he was a visiting scholar at the Department of Chemical and Biological Engineering, University of Wisconsin-Madison, USA. His research interests include nonlinear control, estimation and system analysis, numerical optimization and embedded model predictive
References (15)
- et al.
Modeling and optimization with Optimica and JModelica.org—languages and tools for solving large-scale dynamic optimization problem
Computers and Chemical Engineering
(2010) - et al.
Economic model predictive control with self-tuning terminal cost
European Journal of Control
(2013) Optimizing process economics in model predictive control
(2011)- et al.
Receding horizon cost optimization for overly constrained nonlinear plants
- Angeli, D., Amrit, R., & Rawlings, J.B. (2011). Enforcing convergence in nonlinear economic mpc. In 2011 50th IEEE...
- et al.
On average performance and stability of economic model predictive control
IEEE Transactions on Automatic Control
(2012) - et al.
A lyapunov function for economic optimizing model predictive control
IEEE Transactions on Automatic Control
(2011)
Cited by (18)
Self-stabilizing economic model predictive control without pre-calculated steady-state optima: Stability and robustness
2023, Computers and Chemical EngineeringA new formulation of Economic Model Predictive Control without terminal constraint
2021, AutomaticaCitation Excerpt :In both cases however, it is necessary to be able to limit and in the long term to suppress the movements of actuators in order to achieve a quasi-optimal steady regime as a limit case. As a matter of fact, some approaches include a modification of the economic cost function by adding a tracking term (based on the presumably known optimal equilibrium) (Amrit, Rawlings, & Biegler, 2013; Maree & Imsland, 2016); this can also be interpreted within a multi-objective framework (Zavala, 2015). The discussion above suggests that penalizing the state increment should be effective in deriving a tunable EMPC that addresses the above concerns, as this penalty induces convergence towards an equilibrium.
Hierarchical economic MPC for systems with storage states
2018, AutomaticaCitation Excerpt :In Angeli et al. (2012) the authors showed that strict dissipativity can be achieved by adding a convex regulatory term to the stage cost, while in Maree and Imsland (2014, 2016) the authors propose a combined economic and regulatory MPC algorithm. The dual-objective MPC controller proposed in Maree and Imsland (2014, 2016) consists of both economic and regulatory stage costs, which are dynamically weighted to ensure economically efficient transient performance, whilst also ensuring asymptotic stability of the economic optimal steady state. A similar approach is utilized in this work where an additional term is introduced into the stage cost which penalizes any state not at a steady state.
Economic MPC without terminal constraints: Gradient-correcting end penalties enforce asymptotic stability
2018, Journal of Process ControlCitation Excerpt :In recent years, there has been a growing interest in generalized formulations of Nonlinear Model Predictive Control (NMPC) beyond the classical control tasks of setpoint stabilization and tracking. This includes schemes with purely economic objectives [31,40,10,4,50] and dual formulations [34,5], wherein a combination of tracking and economic objectives is considered. The former approach is termed economic MPC (EMPC) in [40].
Input-to-state stabilizing economic model predictive control of constrained nonlinear systems
2022, Kongzhi yu Juece/Control and DecisionConstrained nonlinear MPC for accelerated tracking piece-wise references and its applications to thermal systems
2022, Control Theory and Technology
Johannes Philippus Maree received Ph.D. degree in Engineering Cybernetics from the Department of Engineering Cybernetics at the Norwegian University of Science and Technology (NTNU) in Trondheim, in 2014. As a part of his Ph.D. studies, he was a visiting scholar at the Department of Chemical and Biological Engineering, University of Wisconsin-Madison, USA. His research interests include nonlinear control, estimation and system analysis, numerical optimization and embedded model predictive control.
Lars Imsland received Ph.D. degree in Electrical Engineering from the Department of Engineering Cybernetics at the Norwegian University of Science and Technology (NTNU) in Trondheim, in 2002. As a part of his Ph.D. studies, he was a visiting scholar at the Institute for Systems Theory in Engineering at the University of Stuttgart, Germany. After his Ph.D. he worked as a postdoctoral researcher at NTNU, a research scientist at SINTEF and as a specialist for Cybernetica AS, before becoming a Professor in Control Engineering at NTNU in 2009. His main research interests are the theory and application of nonlinear and optimizing control and estimation.
- ☆
The material in this paper was partially presented at the 19th IFAC World Congress, August 24–29, 2104, Cape Town, South Africa. This paper was recommended for publication in revised form by Associate Editor David Angeli under the direction of Editor Andrew R. Teel.
- 1
Tel.: +47 73 59 4376.